Feng–Rao decoding of primary codes

Abstract

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [1] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [30] (See also [3]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition in [30] requires the use of differentials which was not needed in [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.

On the evaluation codes given by simple $$\delta $$ δ -sequences

“…According to [45], all these codes have the best known parameters. Codes with Feng-Rao distance equal to 4 and the same remaining parameters can be obtained with the δ-sequence G and they can be efficiently decoded by [27]. Therefore, we have described codes that constitute an improvement with respect to the previously known.…”

“…These papers increase the interest on primary codes. Furthermore, a connection between the Feng-Rao and Andersen-Geil bounds is described in [27], which allows us to decode primary codes and suggests the authors to rename Andersen-Geil bound as Feng-Rao (or order) bound for primary codes. In the sequel, we will use this terminology.…”

“…As an example, we can say that Γ = { (18,9), (6,3), (4,2), (1, 1)} is a δ-sequence determined by Γ * = {45, 15, 10, 3}. Indeed, the pair (m 2 , e 2 ) for Γ * is (3 · 10 − 3, 5) = (27,5) and ⟨a 1 ; a 2 , a 3 ⟩ = ⟨5; 2, 2⟩, so t = 3, y −1 = (0, 1), y 0 = (1, 0) = y t−3 = (A ′ , B ′ ) and (A, B) = 2(1, 0) + (0, 1) = (2, 1). Two δ-sequences in N that determine the same δ-sequence in Z 2 share the same ν-vector.…”

“…An order function defines a filtration of vector spaces contained in its corresponding order domain and, together with an evaluation map, determine two families of error-correcting codes, usually named evaluation and dual families of codes. Lately, these families have been called primary and dual families of (evaluation) codes defined by the pair order function and evaluation map [27].…”

A polynomial time attack against algebraic geometry code based public key cryptosystems

“…This attack will use the concept of error correcting arrays [8, Definition 2.1], [20] or well-behaving sequences [6].…”